On Levi-flat hypersurfaces with prescribed boundary

Abstract

We address the problem of existence and uniqueness of a Levi-flat hypersurface M in ℂn with prescribed compact boundary S for n ≥ 3. The situation for n ≥ 3 differs sharply from the well studied case n = 2. We first establish necessary conditions on S at both complex and CR points, needed for the existence of M. All CR points have to be nonminimal and all complex points have to be "flat". Then, adding a positivity condition at complex points, which is similar to the ellipticity for n = 2 and excluding the possibility of S to contain complex (n - 2)-dimensional submanifolds, we obtain a solution M to the above problem as a projection of a possibly singular Levi-flat hypersurface in ℝ × ℂn. It turns out that S has to be a topological sphere with two complex points and with compact CR orbits, also topological spheres, serving as boundaries of the (possibly singular) complex leaves of M. There are no more global assumptions on S like being contained in the boundary of a strongly pseudoconvex domain, as it was in case n = 2. Furthermore, we show in our situation that any other Levi-flat hypersurface with boundary S must coincide with the constructed solution.